Here is another take on playing with *variational* methods (DFT methods cannot be included in this game):

The total variational energies are obtained at the Def2-TZVPP basis set level:

1. CASSCF(2,5) -1.0440685

2. CASSCF(2,8) -1.0441268

3. CCSD(T) -1.0469931

4. CASSCF(2,5)/MP2 = -1.053945 Hartree.

The last of these, which includes both static and perturbational dynamic correlation energies, is clearly the lowest variationally. One more method, the *multireference coupled cluster* procedure, should give an even lower energy. I don’t have access to such code, but if anyone does, can they report that energy?

An alternate way of calculating frequencies is by numerical differentiation and numerical second derivatives, which uses (in part) different code. Using this procedure one gets:

CASSCF(2,5)/Def2-TZVPP (DOI: 10.14469/hpc/2208) ν -307*i*, 1446, 1818 cm-1

CASSCF(2,8)/Def2-TZVPP (DOI: 10.14469/hpc/2208) ν -306*i*, 1446, 1818 cm-1

This compares with -320i, 1460, 1835 cm-1 obtained at the CCSD(T) level and same basis set. The initial conclusion is that the numerical set of results is more consistent with other methods than the analytical. The analytical frequency code is currently being inspected to try to identify the reason for the discrepancy.

There are in fact other anomalies. In any method which calculates the total energy of a system, one basic requirement is *rotational invariance*, which means that the energy must be identical no matter how the system is oriented for the calculation. One way of checking this invariance is to inspect the values of the six normal modes that represent the translations and rotations in a 3N second derivative matrix. For the numerical derivatives, these emerge as -32.6872 -32.5605 -32.3707 0.1435 0.2795 0.8246 cm-1, which is acceptable for this procedure. For the analytical version these are -0.0002 -0.0002 0.0002 515.2515 518.6620 518.8967 cm-1, with the last three clearly not being close to zero.

The various Hamiltonians do seem to be converging to the conclusion that this species is not a minimum, but a three-fold degenerate transition state for extrusion of H_{2} and two protons.

Yet another Hamiltonian method, the CASSCF (complete active space self-consistent-field) procedure, which variationally optimises all the Slater determinants built from an active orbital space. The previous CCSD and DFT methods were based on just a single variational determinant.

CASSCF(2,5)/Def2-TZVPP (DOI: 10.14469/hpc/2203) ν ~+750, 1472, 1870 cm^{-1}

CASSCF(2,8)/Def2-TZVPP (DOI: 10.14469/hpc/2205) ν ~+600, 1437, 1806 cm^{-1}

This latter constructs 36 determinants based on 2 electrons and an active orbital space of 8, including 7 unoccupied orbitals. This recovers much of the so-called static correlation energy, but less of the dynamic version recovered by e.g. MP2 and CCSD methods. I will investigate these aspects next.

]]>One more basis: ωB97XD/aug-cc-pv6z (a 6-ζ level, DOI: 10.14469/hpc/2188) has ν 97, 1465 and 1780. So the CBS limit using this functional it probably is a real minimum.

I would add this is probably the largest basis set I have ever used in anger! I am also intrigued that the CBS limit is so difficult to reach!

]]>1. Def2-TZVPP (DOI: 10.14469/hpc/2181 ), ν +303, 1462, 1784.

2. Def2-QZVPP (DOI: 10.14469/hpc/2182), ν +173, 1466, 1779.

So probably at the CBS (complete basis set) limit, its probably a true minimum, but still worth testing!

But what about the Hamiltonian? The definitive test is CCSD(T).

3. CCSD(T)/Def2-TZVPP (DOI: 10.14469/hpc/2183), ν ~ -320*i*, 1460, 1835

4. CCSD(T)/Def2-QZVPP (DOI: 10.14469/hpc/2186), ν -325*i*, 1459, 1822.

So it seems its not the basis set but the Hamiltonian method that is sensitive. Such a big difference is a bit of a surprise; you would think that with just two electrons, a standard DFT method should not disagree so much with a coupled cluster method?

My purpose anyway was to explore unusual topologies in electron densities, and less to probe whether H_{4}^{2+} was a minimum or not.

http://onlinelibrary.wiley.com/doi/10.1002/jcc.540140305/full

(Popular basis sets using the standard p exponent suggest (erroneously) that the Td geometry is a minimum) ]]>

M.N. Glukhovtsev, P.v.R. Schleyer, K. Lammertsma, “Can the H2+4 dication exist?” Chem. Phys. Lett., 1993, 209, 207-210. DOI: 10.1016/0009-2614(93)80094-6

(Answer: No) ]]>